… We study the limit cycles of generalized Kukles polynomial differential systems using
averaging theory of first and second order. … Here we consider a very particular case of the
sixteenth Hilbert problem; we want to study the upper bound of the generalized Kukles
polynomial differential system (1) x ̈ = − y , y ̇ = Q ( x , y ) , where Q ( x , y ) is a polynomial
with real coefficients of degree n . This system was introduced by Kukles in [8], giving
necessary and sufficient conditions in order that the system (2) x ̈ = − y , y ̇ = x + a 0 y + a …